1Cademy - Graphing $$y \leq -4x$$

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Graphing a Linear Inequality in Two Variables

Example

Graphing $y \leq -4x$

To graph the inequality $y \leq -4x$ , apply the three-step graphing procedure:

Step 1 — Graph the boundary line. Replace the $\leq$ symbol with $=$ to obtain the boundary line $y = -4x$ . This equation is already in slope–intercept form with slope $m = -4$ and y-intercept $b = 0$ , so the line passes through the origin. Because the inequality uses $\leq$ (non-strict), draw the boundary line as a solid line.

Step 2 — Test a point. Since the boundary line passes through the origin, the point $(0, 0)$ cannot be used as a test point. Instead, choose $(1, 0)$ , which is clearly not on the line:

$0 \stackrel{?}{\leq} -4(1)$

$0 \stackrel{?}{\leq} -4$

$0 \not\leq -4$

The statement is false, so $(1, 0)$ is not a solution of the inequality.

Step 3 — Shade the correct side. Because the test point $(1, 0)$ is not a solution, shade the half-plane on the opposite side — the region to the left of the boundary line.

This example illustrates the special case where the boundary line passes through the origin. Because $(0, 0)$ lies on the line, it cannot serve as a test point, and a different point such as $(1, 0)$ must be selected instead.

Updated 2026-05-03

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